1. © 2010 W. W. Norton & Company, Inc. 3 Preferences

2. © 2010 W. W. Norton & Company, Inc. 2 Rationality in Economics u Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives. u So to model choice we must model decisionmakers’ preferences.

3. © 2010 W. W. Norton & Company, Inc. 3 Preference Relations u Comparing two different consumption bundles, x and y: –strict preference: x is more preferred than is y. –weak preference: x is as at least as preferred as is y. –indifference: x is exactly as preferred as is y.

4. © 2010 W. W. Norton & Company, Inc. 4 Preference Relations u Strict preference, weak preference and indifference are all preference relations. u Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.

5. © 2010 W. W. Norton & Company, Inc. 5 Preference Relations u denotes strict preference; x y means that bundle x is preferred strictly to bundle y.  

6. © 2010 W. W. Norton & Company, Inc. 6 Preference Relations u denotes strict preference; x y means bundle x is preferred strictly to bundle y.   denotes indifference; x  y means x and y are equally preferred.  

7. © 2010 W. W. Norton & Company, Inc. 7 Preference Relations u denotes strict preference so x y means that bundle x is preferred strictly to bundle y.   denotes indifference; x  y means x and y are equally preferred.  denotes weak preference; x y means x is preferred at least as much as is y. ~  ~   

8. © 2010 W. W. Norton & Company, Inc. 8 Preference Relations  x y and y x imply x  y. ~  ~ 

9. © 2010 W. W. Norton & Company, Inc. 9 Preference Relations  x y and y x imply x  y. u x y and (not y x) imply x y. ~  ~  ~  ~  

10. © 2010 W. W. Norton & Company, Inc. 10 Assumptions about Preference Relations u Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x. ~  ~ 

11. © 2010 W. W. Norton & Company, Inc. 11 Assumptions about Preference Relations u Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x. ~ 

12. © 2010 W. W. Norton & Company, Inc. 12 Assumptions about Preference Relations u Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z. ~  ~  ~ 

13. © 2010 W. W. Norton & Company, Inc. 13 Indifference Curves  Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y  x’. u Since an indifference “curve” is not always a curve a better name might be an indifference “set”.

14. © 2010 W. W. Norton & Company, Inc. 14 Indifference Curves x2x2x2x2 x1x1x1x1 x” x”’ x’  x”  x”’ x’

15. © 2010 W. W. Norton & Company, Inc. 15 Indifference Curves x2x2x2x2 x1x1x1x1 zxy z x y  x y z

16. © 2010 W. W. Norton & Company, Inc. 16 Indifference Curves x2x2 x1x1 x All bundles in I 1 are strictly preferred to all in I 2. y z All bundles in I 2 are strictly preferred to all in I 3. I1I1 I2I2 I3I3

17. © 2010 W. W. Norton & Company, Inc. 17 Indifference Curves x2x2 x1x1 I(x’) x I(x) WP(x), the set of bundles weakly preferred to x.

18. © 2010 W. W. Norton & Company, Inc. 18 Indifference Curves x2x2 x1x1 WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x). x I(x)

19. © 2010 W. W. Norton & Company, Inc. 19 Indifference Curves x2x2 x1x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x)

20. © 2010 W. W. Norton & Company, Inc. 20 Indifference Curves Cannot Intersect x2x2x2x2 x1x1x1x1 x y z I1I1I1I1 I2I2 From I 1, x  y. From I 2, x  z. Therefore y  z.

21. © 2010 W. W. Norton & Company, Inc. 21 Indifference Curves Cannot Intersect x2x2x2x2 x1x1x1x1 x y z I1I1I1I1 I2I2 From I 1, x  y. From I 2, x  z. Therefore y  z. But from I 1 and I 2 we see y z, a contradiction. 

22. © 2010 W. W. Norton & Company, Inc. 22 Slopes of Indifference Curves u When more of a commodity is always preferred, the commodity is a good. u If every commodity is a good then indifference curves are negatively sloped.

23. © 2010 W. W. Norton & Company, Inc. 23 Slopes of Indifference Curves Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve.

24. © 2010 W. W. Norton & Company, Inc. 24 Slopes of Indifference Curves u If less of a commodity is always preferred then the commodity is a bad.

25. © 2010 W. W. Norton & Company, Inc. 25 Slopes of Indifference Curves Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve.

26. © 2010 W. W. Norton & Company, Inc. 26 Extreme Cases of Indifference Curves; Perfect Substitutes u If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount of the two commodities in bundles determines their preference rank-order.

27. © 2010 W. W. Norton & Company, Inc. 27 Extreme Cases of Indifference Curves; Perfect Substitutes x2x2x2x2 x1x1x1x1 8 8 15 15 Slopes are constant at - 1. I2I2 I1I1 Bundles in I 2 all have a total of 15 units and are strictly preferred to all bundles in I 1, which have a total of only 8 units in them.

28. © 2010 W. W. Norton & Company, Inc. 28 Extreme Cases of Indifference Curves; Perfect Complements u If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements and only the number of pairs of units of the two commodities determines the preference rank-order of bundles.

29. © 2010 W. W. Norton & Company, Inc. 29 Extreme Cases of Indifference Curves; Perfect Complements x2x2x2x2 x1x1x1x1 I1I1 45 o 5 9 59 Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred.

30. © 2010 W. W. Norton & Company, Inc. 30 Extreme Cases of Indifference Curves; Perfect Complements x2x2x2x2 x1x1x1x1 I2I2 I1I1 45 o 5 9 59 Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9) which contains 9 pairs.

31. © 2010 W. W. Norton & Company, Inc. 31 Preferences Exhibiting Satiation u A bundle strictly preferred to any other is a satiation point or a bliss point. u What do indifference curves look like for preferences exhibiting satiation?

32. © 2010 W. W. Norton & Company, Inc. 32 Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Satiation (bliss) point

33. © 2010 W. W. Norton & Company, Inc. 33 Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point

34. © 2010 W. W. Norton & Company, Inc. 34 Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point

35. © 2010 W. W. Norton & Company, Inc. 35 Indifference Curves for Discrete Commodities u A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. u A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators.

36. © 2010 W. W. Norton & Company, Inc. 36 Indifference Curves for Discrete Commodities u Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like?

37. © 2010 W. W. Norton & Company, Inc. 37 Indifference Curves With a Discrete Good Gas- oline Aircraft 01234 Indifference “curves” are collections of discrete points.

38. © 2010 W. W. Norton & Company, Inc. 38 Well-Behaved Preferences u A preference relation is “well- behaved” if it is –monotonic and convex. u Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good).

39. © 2010 W. W. Norton & Company, Inc. 39 Well-Behaved Preferences u Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of the bundles x and y is z = (0.5)x + (0.5)y. z is at least as preferred as x or y.

40. © 2010 W. W. Norton & Company, Inc. 40 Well-Behaved Preferences -- Convexity. x2x2x2x2 y2y2y2y2 x 2 +y 2 2 x1x1x1x1 y1y1y1y1 x 1 +y 1 2 x y z = x+y 2 is strictly preferred to both x and y.

41. © 2010 W. W. Norton & Company, Inc. 41 Well-Behaved Preferences -- Convexity. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 x y z =(tx 1 +(1-t)y 1, tx 2 +(1-t)y 2 ) is preferred to x and y for all 0 < t < 1.

42. © 2010 W. W. Norton & Company, Inc. 42 Well-Behaved Preferences -- Convexity. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 x y Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. z

43. © 2010 W. W. Norton & Company, Inc. 43 Well-Behaved Preferences -- Weak Convexity. x’ y’ z’ Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle. x z y

44. © 2010 W. W. Norton & Company, Inc. 44 Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

45. © 2010 W. W. Norton & Company, Inc. 45 More Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

46. © 2010 W. W. Norton & Company, Inc. 46 Slopes of Indifference Curves u The slope of an indifference curve is its marginal rate-of-substitution (MRS). u How can a MRS be calculated?

47. © 2010 W. W. Norton & Company, Inc. 47 Marginal Rate of Substitution x2x2x2x2 x1x1x1x1 x’ MRS at x’ is the slope of the indifference curve at x’

48. © 2010 W. W. Norton & Company, Inc. 48 Marginal Rate of Substitution x2x2x2x2 x1x1x1x1 MRS at x’ is lim {x 2 /x 1 } x 1 0 = dx 2 /dx 1 at x’ MRS at x’ is lim {  x 2 /  x 1 }  x 1 0 = dx 2 /dx 1 at x’ x2x2x2x2 x1x1x1x1 x’

49. © 2010 W. W. Norton & Company, Inc. 49 Marginal Rate of Substitution x2x2x2x2 x1x1 dx 2 dx 1 dx 2 = MRS dx 1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. dx 2 = MRS  dx 1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x’

50. © 2010 W. W. Norton & Company, Inc. 50 MRS & Ind. Curve Properties Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve MRS < 0.

51. © 2010 W. W. Norton & Company, Inc. 51 MRS & Ind. Curve Properties Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS > 0.

52. © 2010 W. W. Norton & Company, Inc. 52 MRS & Ind. Curve Properties Good 2 Good 1 MRS = - 5 MRS = - 0.5 MRS always increases with x 1 (becomes less negative) if and only if preferences are strictly convex.

53. © 2010 W. W. Norton & Company, Inc. 53 MRS & Ind. Curve Properties x1x1x1x1 x2x2x2x2 MRS = - 0.5 MRS = - 5 MRS decreases (becomes more negative) as x 1 increases nonconvex preferences

54. © 2010 W. W. Norton & Company, Inc. 54 MRS & Ind. Curve Properties x2x2x2x2 x1x1x1x1 MRS = - 0.5 MRS = - 1 MRS = - 2 MRS is not always increasing as x 1 increases nonconvex preferences.